Generalizations of Tucker–Fan–Shashkin Lemmas

Oleg R. Musin
This research is partially
supported by the NSF Grant DMS-1400876 and the RFBR Grant
15-01-99563School of Mathematical and Statistical Sciences University
of Texas Rio Grande Valley, One West University Boulevard Brownsville TX 78520 USA
oleg.musin@utrgv.edu

Tucker and Ky Fan’s lemma are combinatorial analogs of the
Borsuk–Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a
version of Fan’s lemma, which is a combinatorial analog of the odd
mapping theorem (OMT). We consider generalizations of these lemmas
for BUT–manifolds, i.e. for manifolds that satisfy BUT. Proofs rely
on a generalization of the OMT and on a lemma about the doubling of
manifolds with boundaries that are BUT–manifolds.

Throughout this paper the symbol ${\mathbb{R}}^{d}$ denotes the
Euclidean space of dimension $d$. We denote by ${\mathbb{B}}^{d}$ the
$d$-dimensional unit ball and by ${\mathbb{S}}^{d}$ the
$d$-dimensional unit sphere. If we consider ${\mathbb{S}}^{d}$ as the
set of unit vectors $x$ in ${\mathbb{R}}^{d+1}$, then points $x$ and
$-x$ are called antipodal and the symmetry given by the
mapping $x\to-x$ is called the antipodality on ${\mathbb{S}}^{d}$.

Let $T$ be a triangulation of the $d$-dimensional ball ${\mathbb{B}}^{d}$. We call $T$ antipodally symmetric on the boundary if
the set of simplices of $T$ contained in the boundary of ${\mathbb{B}}^{d}={\mathbb{S}}^{d-1}$ is an antipodally symmetric triangulation
of ${\mathbb{S}}^{d-1}$; that is if $s\subset{\mathbb{S}}^{d-1}$ is
a simplex of $T$, then $-s$ is also a simplex of $T$.

Tucker’s Lemma[Tucker1945]Let $T$ be a triangulation
of ${\mathbb{B}}^{d}$ that is antipodally symmetric on the boundary.
Let

$\displaystyle L:V(T)\to\{+1,-1,+2,-2,\ldots,+d,-d\}$

be a labelling of the vertices of $T$ that is antipodal
(i. e. $L(-v)=-L(v)$) for every vertex $v$ on the
boundary. Then there exists an edge in $T$ that is complementary, i.e., its two vertices are labelled by opposite
numbers.

Ky Fan’s LemmaLet $T$ be a centrally
symmetric triangulation of the sphere ${\mathbb{S}}^{d}$. Suppose that each vertex $v$ of $T$ is assigned a label
$L(v)$ from $\{\pm 1,\pm 2,\ldots,\pm n\}$ in such a way
that $L(-v)=-L(v)$. Suppose this labelling does not have
complementary edges. Then there are an odd number of $d$-simplices of $T$ whose labels are of the form
$\{k_{0},-k_{1},k_{2},\ldots,(-1)^{d}k_{d}\}$, where $1\leq k_{0}<k_{1}<\cdots<k_{d}\leq n$. In particular, $n\geq d+1$.

In the 1990s, Yu. A. Shashkin published several works related to
discrete versions of classic fixed point theorems
([Shashkin1991]; [Shashkin1994]; [Shashkin1996a]; [Shashkin1996b]; [Shashkin1999]). In
[Shashkin1996b] he proved the following theorem:

Shashkin’s LemmaLet $T$ be a
triangulation of a planar polygon that is antipodally symmetric on
the boundary. Let

$\displaystyle L:V(T)\to\{+1,-1,+2,-2,+3,-3\}$

be a labelling of the vertices of $T$ that satisfies
$L(-v)=-L(v)$ for every vertex $v$ on the boundary.
Suppose that this labelling does not have complementary edges. Then
for any numbers $a,b,c$, where $|a|=1,\;|b|=2,\;|c|=3$,
the total number of triangles in $T$ with labels
$(a,b,c)$ and $(-a,-b,-c)$ is odd.

Figure 2: Illustration of Shashkin’s lemma

Remark.

In other words, Shashkin proved that if $(a,b,c)=(1,2,3),\,(1,-2,3),{}(1,2,-3)$ and $(1,-2,-3)$, then the number of triangles
with labels $(a,b,c)$ or $(-a,-b,-c)$ is odd. Denote this number by
${\rm SN}(a,b,c)$. Then in Fig. 2 we have

Note that this result was published only in Russian and only for
two–dimensional case. Moreover, Shashkin attributes this theorem to
[Fan1952].

Actually, Shashkin’s lemma can be derived from Ky Fan’s lemma for
$n=d+1$. However. Shashkin’s proof is different and relies on the
odd mapping theorem (OMT). In fact, this lemma is a discrete
version of the OMT. That is why we distinguish this result as Shashkin’s lemma.

be an antipodal labelling of $T$. Suppose that this
labelling does not have complementary edges. Then for any set of
labels
$\Lambda:=\{\ell_{1},\ell_{2},\ldots,\ell_{d+1}\}\subset\Pi_{d+1}$ with $|\ell_{i}|=i$ for all $i$, the number of $d$–simplices in $T$ that are labelled by $\Lambda$ is
odd.

In [Musin2012] we invented BUT (Borsuk–Ulam Type) – manifolds.
Theorems 3.1–3.4 in this paper extend Tucker’s and Shashkin’s
lemmas for BUT–manifolds. Namely, Theorem 3.1 and Theorem 3.2 are
extensions of the spherical Tucker and Shahskin lemmas, where
${\mathbb{S}}^{d}$ is substituted by a BUT–manifold. Theorems 3.3 and
3.4 are extensions of the original Tucker and Shashkin lemmas,
where ${\mathbb{B}}^{d}$ is substituted by a manifold $M$ with boundary
$\partial M$ that is a BUT–manifold.

Our proof of Theorem 3.2 is relies on a generalization of the odd
mapping theorem for BUT–manifolds:

Theorem 2.1.

Let $(M_{1},A_{1})$ and $(M_{2},A_{2})$ be BUT–manifolds. Then any odd
continuous mapping $f:M_{1}\to M_{2}$ has odd degree.

Theorems 3.3 and 3.4 follow from Theorems
3.1 and 3.2 by using Lemma 3.1, which is about
the doubling of manifolds with boundaries that are BUT–manifolds.

In Section 4 we extend for BUT–manifolds Shaskin’s proof of two
Tucker’s theorems about covering families from [Tucker1945].
Actually, these theorems are corollaries of Theorem 3.2.

We say that a mapping $f:{\mathbb{S}}^{d}\to{\mathbb{S}}^{d}$ is odd or
antipodal if $f(-x)=-f(x)$ for all $x\in{\mathbb{S}}^{d}$. If $f$
is a continuous mapping, then $\deg{f}$ (the degree of $f$) is well
defined.

Let $f:M_{1}\to M_{2}$ be a continuous map between two closed manifolds
$M_{1}$ and $M_{2}$ of the same dimension. The degree is a number that
represents the amount of times that the domain manifold wraps around
the range manifold under the mapping. Then $\operatorname{deg_{2}}2(f)$ (the degree
modulo 2) is 1 if this number is odd and 0 otherwise. It is well
known that $\operatorname{deg_{2}}2(f)$ of a continuous map $f$ is a homotopy
invariant [see ([Milnor1969])].

[Shashkin1996b] (see also [[Matoušek2003], Proposition 2.4.1])
gives a proof of this theorem for simplicial mappings $f:{\mathbb{S}}^{d}\to{\mathbb{S}}^{d}$. [Conner and Floyd1960] considered Theorem
2.1 for a wide class of spaces. Here we extend the odd mapping
theorem for BUT–manifolds. In our paper [Musin2012], we extended the
Borsuk–Ulam theorem for manifolds.

Let $M$ be a connected compact PL (piece-wise linear)
$d$-dimensional manifold without boundary with a free simplicial
involution $A:M\to M$, i. e. $A^{2}(x)=A(A(x))=x$ and $A(x)\neq x$. We
say that a pair $(M,A)$ is a BUT (Borsuk-Ulam Type) manifold
if for any continuous $g:M\to{\mathbb{R}}^{d}$ there is a point $x\in M$ such that $g(A(x))=g(x)$. Equivalently, if a continuous map $f:M\to{\mathbb{R}}^{d}$ is antipodal, i.e. $f(A(x))=-f(x)$, then the
set of zeros $Z_{f}:=f^{-1}(0)$ is not empty.

In [Musin2012], we found several equivalent necessary and sufficient
conditions for manifolds to be BUT. In particular,

$M$ is a $d$–dimensional BUT–manifold if and only if $M$
admits an antipodal continuous transversal to zeros mapping $h:M\to{\mathbb{R}}^{d}$ with $|Z_{h}|=2\;\;(\mathop{{\rm mod}}4)$.

A continuous mapping $h:M\to{\mathbb{R}}^{d}$ is called transversal
to zero if there is an open set $U$ in ${\mathbb{R}}^{d}$ such that $U$
contains 0, $U$ is homeomorphic to the open $d$-ball and
$h^{-1}(U)$ consists of a finite number open sets in $M$ that are
homeomorphic to open $d$-balls.

The class of BUT–manifolds is sufficiently large. It is clear that
$({\mathbb{S}}^{d},A)$ with $A(x)=-x$ is a BUT-manifold. Suppose that
$M$ can be represented as a connected sum $N\#N$, where $N$ is a
closed PL manifold. Then $M$ admits a free involution. Indeed, $M$
can be “centrally symmetrically” embedded to ${\mathbb{R}}^{k}$, for
some $k$, and the antipodal symmetry $x\to-x$ in ${\mathbb{R}}^{k}$
implies a free involution $A:M\to M$ [[Musin2012], Corollary 1]. For
instance, orientable two-dimensional manifolds $M^{2}_{g}$ with even
genus $g$ and non-orientable manifolds $P^{2}_{m}$ with even $m$, where
$m$ is the number of Möbius bands, are BUT-manifolds.

Let $M_{i},\;i=1,2$, be a manifold with a free involution $A_{i}$. We
say that a mapping $f:M_{1}\to M_{2}$ is antipodal (or odd,
or equivariant) if $f(A_{1}(x))=A_{2}(f(x))$ for all $x\in M_{1}$.

Theorem 2.1.

Since $(M_{2},A_{2})$ is BUT, there is a continuous antipodal
transversal to zeros mapping $g:M_{2}\to{\mathbb{R}}^{d}$ with
$|Z_{g}|=4m_{2}+2$ [[Musin2012], Theorem 2].

Let $h:=g\circ f$. Then $h:M_{1}\to{\mathbb{R}}^{d}$ is continuous and
antipodal. Since the degree of a mapping is a homotopy
invariant, without loss of generality we may assume that $h$ is a
transversal to zero mapping (see [Musin2012], [Lemma 3]). Therefore
$|Z_{h}|=4m_{1}+2$. On the other hand,

In our papers [Musin2012]; [Musin2015], [Musin and Volovikov2015] are considered extensions of
Tucker’s lemma. Here we consider generalizations of Tucker’s and
Shashkin’s lemmas for manifolds with and without boundaries.

Let $T$ be an antipodally symmetric (or antipodal)
triangulation of a BUT–manifold $(M,A)$. This means that $A:T\to T$
sends simplices to simplices. Denote by $\Pi_{n}$ the set of labels
$\{+1,-1,+2,-2,\ldots,+n,-n\}$ and let $L:V(T)\to\Pi_{n}$ be a
labeling of $T$. We say that this labelling is antipodal if
$L(A(v))=-L(v)$. An edge $uv$ in $T$ is called complementary
if $L(u)=-L(v)$.

Theorem 3.1.

([Musin2015], Theorem 4.1) Let $(M,A)$ be a
$d$-dimensional BUT–manifold. Let $T$ be an antipodal triangulation
of $M$. Then for any antipodal labelling $L:V(T)\to\Pi_{d}$ there
exists a complementary edge.

Any antipodal labelling $L:V(T)\to\Pi_{n}$ of an antipodally
symmetric triangulation $T$ of $M$ defines a simplicial map
$f_{L}:T\to{\mathbb{R}}^{n}$. Let
$\{e_{1},-e_{1},e_{2},-e_{2},\ldots,e_{n},-e_{n}\}$ be the standard orthonormal
basis in ${\mathbb{R}}^{n}$. For $v\in V(T)$, set $f_{L}(v):=e_{i}$ if
$L(v)=i$ and $f_{L}(v):=-e_{i}$ if $L(v)=-i$. Since $f_{L}$ is defined on
$V(T)$, it defines a simplicial mapping $f_{L}:T\to{\mathbb{R}}^{n}$
(see details in [[Matoušek2003], Sect. 2.3].)

The following theorem is a version of Shashkin’s lemma for manifolds
without boundary.

Theorem 3.2.

Let $(M,A)$ be a $d$-dimensional BUT–manifold. Let $T$ be an
antipodally symmetric triangulation of $M$. Let $L:V(T)\to\Pi_{d+1}$ be an antipodal labelling of $T$. Suppose that this
labelling does not have complementary edges. Then for any set of
labels
$\Lambda:=\{\ell_{1},\ell_{2},\ldots,\ell_{d+1}\}\subset\Pi_{d+1}$ with
$|\ell_{i}|=i$ for all $i$, the number of $d$–simplices in $T$ that
are labelled by $\Lambda$ is odd.

Since $L$ has no complimentary edges, $f_{L}:T\to{\mathbb{R}}^{d+1}$ is
an antipodal mapping of $M$ to the boundary of the crosspolytope
$C^{d+1}$ that is the convex hull ${\rm conv}{\{e_{1},-e_{1},\ldots,e_{d+1},-e_{d+1}\}}.$ Note that $\partial C^{d+1}$ is a simplicial sphere ${\mathbb{S}}^{d}$, which is a
BUT-manifold. Therefore, Theorem 2.2 implies that the number of
preimages of the simplex in $\partial C^{d+1}$ with indexes from
$\Lambda$ is odd. It completes the proof.
$\square$

Remark.

Theorem 3.1 can be proved using the same arguments. Indeed, suppose
that $L:V(T)\to\Pi_{d}$ has no complementary edges. Then $f_{L}$ sends
$M$ to $\partial C^{d}$. Since $\dim{\partial C^{d}}=d-1$,
$\deg{f_{L}}=0$. This contradicts Theorem 2.1.

Now we extend Tucker’s and Shashkin’s lemmas for the case when $M$
is a manifold with boundary that is a BUT–manifold. But first,
prove that there exists a “double” of $M$ that is a BUT-manifold.

Lemma 3.1.

Let $M$ be a compact PL manifold with boundary $\partial M$. Suppose
$(\partial M,A)$ is a BUT–manifold. Then there is a BUT–manifold
$(\tilde{M},\tilde{A})$ and a submanifold $N$ in $\tilde{M}$ such that
$N\simeq M$, $\tilde{A}|_{\partial N}\simeq A$, $(N{\setminus}\partial N)\cap\tilde{A}(N{\setminus}\partial N)=\emptyset$ and

First we prove the following statement:
Let
$X$
be a finite simplicial complex. Let
$Y$
be a subcomplex
of
$X$
with a free involution
$A:Y\to Y$
. Then there is a simplicial
embedding
$F$
of
$X$
into
${\mathbb{R}}^{q}_{+}:=\{(x_{1},\ldots,x_{q})\in{\mathbb{R}}^{q}:x_{1}\geq 0\}$
, where
$q$
is sufficiently large, such that
$Y$
is centrally symmetrically embedded in
${\mathbb{R}}^{q}$
, i.e.
$F(A(y))=-F(y)$
for all
$y\in Y$
, and
$X{\setminus}Y$
is mapped into
the interior of
${\mathbb{R}}^{q}_{+}$
.

Indeed, let
$v_{1},v_{-1},\ldots,v_{m},v_{-m}$
denote vertices of
$Y$
such that
$A(v_{k})=v_{-k}$
. Let
$\{v_{m+1},\ldots,v_{n}\}$
be the set
of vertices of
$X{\setminus}Y$
.

Denote by
$C^{n}$
the
$n$
–dimensional crosspolytope that is the
boundary of convex hull

$\displaystyle{\rm conv}{\{e_{1},-e_{1},\ldots,e_{n},-e_{n}\}}$

of the vectors of the standard orthonormal basis and their negatives.

Now define an embedding
$F:X\to C^{n}$
. Let
$F(v_{k}):=e_{k}$
,
$F(v_{-k}):=e_{-k}$
, where
$1\leq k\leq m$
,
$F(v_{k}):=e_{k}$
, and
$k=m+1,\ldots,n$
. Since
$F$
is defined for all of the vertices of
$X$
, it uniquely defines a simplicial (piecewise linear) mapping
$F:X\to C^{n}\subset{\mathbb{R}}^{n}$
. Then

Let
$X=M$
and
$Y=\partial M$
. Then it follows from 1 that there is an embedding
$F:M\to{\mathbb{R}}^{q}_{+}$
with
$F(\partial M)\subset{\mathbb{R}}^{q}$
and
$F(A(y))=-F(y)$
for all
$y\in\partial M$
, where
$q=n-1$
.
Let

It is clear that
$\tilde{M}\simeq(N{\setminus}\partial N)\cup\partial N\cup\tilde{A}(N{\setminus
}\partial N),$
where
$N:=F(M).$

3.

Let us prove that
$(\tilde{M},\tilde{A})$
is BUT. Indeed, since
$(\partial M,A)$
is BUT, there is a continuous antipodal transversal to zeros mapping
$g:\partial M\simeq\partial N\to{\mathbb{R}}^{d-1}$
with
$|Z_{g}|=4m+2$
, where
$d:=\dim{M}.$
We extend this mapping to
$h:\tilde{M}\to{\mathbb{R}}^{d}$
with
$h|_{\partial N}=g$
and
$|Z_{h}|=|Z_{g}|=4m+2$
.

Let $v=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$ be a vertex of $\tilde{M}$.
If $v\in\partial N$, then put

Then $h:\tilde{M}\to{\mathbb{R}}^{d}$ is an antipodal transversal to
zeros mapping and $h^{-1}(0)=g^{-1}(0)$.
$\square$

Theorem 3.3.

Let $M$ be a $d$–dimensional compact PL manifold with boundary
$\partial M$. Suppose $(\partial M,A)$ is a BUT–manifold. Let $T$
be a triangulation of $M$ that antipodally symmetric on $\partial M$. Let $L:V(T)\to\Pi_{d}$ be a labelling of $T$ that is antipodal
on the boundary. Then there is a complementary edge in $T$.

Theorem 3.4.

Let $M$ be a $d$–dimensional compact PL manifold with boundary
$\partial M$. Suppose $(\partial M,A)$ is a BUT–manifold. Let $T$
be a triangulation of $M$ that antipodally symmetric on $\partial M$. Let $L:V(T)\to\Pi_{d+1}$ be a labelling of $T$ that is
antipodal on the boundary and has no complementary edges. Then for
any set of labels
$\Lambda:=\{\ell_{1},\ell_{2},\ldots,\ell_{d+1}\}\subset\Pi_{d+1}$ with
$|\ell_{i}|=i$ for all $i$, the number of $d$–simplices in $T$ that
are labelled by $\Lambda$ or $(-\Lambda)$ is odd.

By Lemma 3.1 there is a BUT–manifold $(\tilde{M},\tilde{A})$ that is the double of $M$. We can extend $T$ and $L$ from $M$
to an antipodal triangulation $\tilde{T}:=T\cup\tilde{A}(T)$ of
$\tilde{M}$ and an antipodal labelling $\tilde{L}:V(\tilde{T})\to\Pi_{n}$, where $n=d$ in Theorem 3.3 and $n=d+1$ in Theorem
3.4, such that $\tilde{L}|_{T}=L$.

In this section we consider two Tucker’s theorems about covering
families. Note that [Tucker1945] obtained these theorem only
for ${\mathbb{S}}^{2}$. [Bacon1966] proved that statements in
Theorems 4.1 and 4.2 are equivalent to the
Borsuk–Ulam theorem for normal topological spaces $X$ with free
continuous involutions $A:X\to X$. [See also Theorem 2.1
in our paper ([Musin and Volovikov2015])]. Actually, these theorems can be proved
from properties of Schwarz’s genus ([Svarc1966]) or Yang’s
cohomological index ([Karasev2009]; [Musin and Volovikov2015]).

For the two–dimensional case in the book [Shashkin1999]
Shashkin derives Tucker’s theorems from his lemma. Here we extend
his proof for BUT–manifolds of all dimensions.

Theorem 4.1.

Let $(M,A)$ be a $d$-dimensional BUT–manifold. Consider a family of
closed sets $\{B_{i},B_{-i}\},\,i=1,\ldots,d+1$, where
$B_{-i}:=A(B_{i})$, is such that $B_{i}\cap B_{-i}=\emptyset$ for all
$i$. If this family covers $M$, then for any set of indices
$\{k_{1},k_{2},\ldots,k_{d+1}\}\subset\Pi_{d+1}$ with $|k_{i}|=i$ for all
$i$, the intersection of all $B_{k_{i}}$ is not empty.

Note that any PL manifold admits a metric. For a triangulation $T$
of $M$, the norm of $T$, denoted by $|T|$, is the diameter of the
largest simplex in $T$.

Let $T_{1},T_{2},\ldots$ be a sequence of antipodal triangulations of
$M$ such that $|T_{i}|\to 0$. Now for all $i$ define an antipodal
labelling $L_{i}:V(T_{i})\to\Pi_{d+1}$. For every $v\in V(T_{i})\subset M$ set

Then $L_{i}$ satisfies the assumptions in Theorem 3.2 and $T_{i}$
contains a simplex $s_{i}$ with labels
$\{k_{1},k_{2},\ldots,k_{d+1}\}\subset\Pi_{d+1}$.

Since $M$ is compact and $|s_{i}|\to 0$, the sequence $\{s_{i}\}$
contains a converging subsequence $P$ with limit $w\in M$. Then for
$s_{i}\in P$ we have $V(s_{i})\to w$.

By assumption, all $B_{k}$ are closed sets. Therefore $w\in B_{k_{j}}$
for all $j=1,\ldots,d+1$, and thus $w\in\cap_{j}{B_{k_{j}}}$.
$\square$

Theorem 4.2.

Let $(M,A)$ be a $d$-dimensional BUT–manifold. Suppose that $M$ is
covered by a family $\mathcal{F}$ of $d+2$ closed subsets
$C_{1},\ldots,C_{d+2}$. Suppose that all $C_{i}$ have no antipodal pairs
$(x,A(x))$, in other words, $C_{i}\cap A(C_{i})=\emptyset$. Let
$0<k<d+2$. Then any $k$ subsets from $\mathcal{F}$ intersect and
there is a point $x$ in this intersection such that $A(x)$ belongs
to the intersection of the remaining $(d+2-k)$ subsets in $\mathcal{F}$.

On the other hand, $B_{i}\subset C_{i}$ and $B_{-i}\subset C_{-i}$,
hence $B_{i}\cap B_{-i}=\emptyset$. Therefore, the family of subsets
$\{B_{i}\}$ satisfies the assumptions of Theorem 4.1. It follows
that